Remark 5.10 With the hypothesis in Lemma REF , one may follow the similar argument in the proof of [[1]}, Lemma 3.9] to find a new family \(\widetilde{\varphi }\)  (with some \(\mathbb {C}^*\) -action on \(\varphi \) ) such that \(\mathrm {Supp}(\mathrm {coker}(\widetilde{\varphi }|_{\widetilde{\eta }_{0}}))\)  is away from the divisors \(\mathcal {D}_{\pm }\)  (it is equivalent to the admissibility of \(\mathrm {coker}(\widetilde{\varphi }|_{\widetilde{\eta }_{0}})\)  by Remark REF ) possibly after a finite base change \(\widetilde{S}\rightarrow S\)  where \(\widetilde{\eta }_{0}\)  is the closed point of \(\widetilde{S}\) .

